Harmonic Oscillator 1D

U(x)=12mω2x2=12kx2\mathcal{U}(x) = \frac 1 2m\omega^2x^2 = \frac 1 2kx^2
Nn=(2nn!)−1/2(mωπℏ)1/4N_n = (2^nn!)^{-1/2}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \\

Hermites polynom:

Hn(ξ)=(−1)n⋅eξ2⋅dne−ξ2dξnH_n(\xi) = (-1)^n\cdot e^{\xi^2}\cdot \frac{d^n e^{-\xi^2}}{d\xi^n}
Φn(x)=Nn⋅e−mω2ℏx2⋅Hn(mωℏx)\bm\Phi_n(x) = N_n\cdot e^{-\frac{m\omega}{2\hbar}x^2}\cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)
En=ℏω⋅(n+12)E_n = \hbar\omega\cdot\left(n + \frac 1 2\right)

The wave equations can alternatively be written:

un(x)=N(∂∂x−ax)n⋅u0(x)u0(x)=e−ax2/2\begin{gathered} u_n(x) = N\left(\frac{\partial}{\partial x} - ax\right)^n\cdot u_0(x) \\ u_0(x) = e^{-ax^2/2} \end{gathered}

Spherical Symmetric Potential

U(r)=U(r)\mathcal{U}(\bm r) = \mathcal{U}(r)
H=−ℏ2mr2∂∂r[r2∂∂r]+L22mr2+U(r)H = -\frac \hbar{2mr^2} \frac \partial{\partial r} \left[r^2\frac\partial{\partial r}\right] + \frac{L^2}{2mr^2} + \mathcal{U}(r)
Hψnlm(r)=Enlmψnlm(r)H\bm\psi_{nlm}(\bm r) = E_{nlm}\bm\psi_{nlm}(\bm r)
ψnlm(r)=Gnl(r)rΥlm(θ,ϕ)\bm\psi_{nlm}(\bm r) = \frac{G_{nl}(r)}r\Upsilon_l^m(\theta, \phi)

Radial equation:

−ℏ22md2dr2G(r)+[l(l+1)ℏ22mr2+U(r)]G(r)=EG(r)-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}G(r) + \left[\frac{l(l + 1)\hbar^2}{2mr^2} + \mathcal U(r)\right]G(r) = EG(r)

Hydrogen-like Atom

U(r)=−Ze24πϵ0r\mathcal U(r) = -\frac{Ze^2}{4\pi\epsilon_0r}

The Schrödinger equation simplifies to:

[Δ+2Za0r+2mEℏ2]Φ(r)=0\left[\Delta + \frac{2Z}{a_0r} + \frac{2mE}{\hbar^2}\right]\bm\Phi(r) = 0

Radial wave functions of hydrogenic atoms:

nlRnl(r)10R10(r)=2(Za0)3/2e−ρ/220R20(r)=122(Za0)3/2(2−ρ)e−ρ/221R21(r)=126(Za0)3/2ρe−ρ/230R30(r)=193(Za0)3/2(6−6ρ+ρ2)e−ρ/231R31(r)=196(Za0)3/2ρ(4−ρ)e−ρ/232R32(r)=1930(Za0)3/2ρ2e−ρ/2\def\arraystretch{2.5} \begin{array}{ccc} \hline n & l & R_{nl}(r) \\ \hline 1 & 0 & R_{10}(r) = 2\left(\frac Z{a_0}\right)^{3/2}e^{-\rho/2}\\ 2 & 0 & R_{20}(r) = \frac 1{2\sqrt 2}\left(\frac Z{a_0}\right)^{3/2}(2 - \rho)e^{-\rho/2} \\ 2 & 1 & R_{21}(r) = \frac 1{2\sqrt 6}\left(\frac Z{a_0}\right)^{3/2}\rho e^{-\rho/2} \\ 3 & 0 & R_{30}(r) = \frac 1{9\sqrt 3}\left(\frac Z{a_0}\right)^{3/2}\left(6 - 6\rho + \rho^2\right) e^{-\rho/2} \\ 3 & 1 & R_{31}(r) = \frac 1{9\sqrt 6}\left(\frac Z{a_0}\right)^{3/2}\rho(4-\rho) e^{-\rho/2} \\ 3 & 2 & R_{32}(r) = \frac 1{9\sqrt 30}\left(\frac Z{a_0}\right)^{3/2}\rho^2 e^{-\rho/2} \\ \hline \end{array}
E−n=−mZ2e432π2ϵ02ℏ2n2=−Z2ℏ22a02mn2=−13.6Z2n2eVE-n = -\frac{mZ^2e^4}{32\pi^2\epsilon_0^2\hbar^2n^2} = -\frac{Z^2\hbar^2}{2a_0^2mn^2} = -13.6\frac{Z^2}{n^2}\textup{eV}
S(x,t)=ℏ2im[ψ∗⋅∂ψ∂x−ψ∂∂xψ∗]S(x, t) = \frac\hbar {2im}\left[\bm\psi^*\cdot\frac{\partial\bm\psi}{\partial x} - \bm\psi\frac\partial{\partial x} \bm\psi^*\right]